a robust power system stabilizer design

OPTIMAL CONTROL APPLICATIONS & METHODS, VOL. 18, 179—193 (1997)

A ROBUST POWER SYSTEM STABILIZER DESIGN

SHEN CHEN, O. P. MALIK* AND TONGWEN CHEN

Department of Electrical and Computer Engineering, The University of Calgary, 2500 University Drive NW,Calgary, Alta. T2N 1N4, Canada

SUMMARY

Power system stabilizers (PSSs) have been widely used in electric power systems to provide extra dampingfor synchronous generators. An H

=-optimization-based robust control design methodology is presented in

this paper for PSS design. The proposed technique takes into account system uncertainties due toincomplete knowledge of the physical parameters, neglected high-frequency dynamics and invalid assump-tions made in the model formulation process. It also focuses on power system stabilizer design usinga normal design model with an uncertainty description which represents the possible perturbation ofa synchronous generator around its nominal operating point. The proposed excitation controller enables thepower system to be stable over a wide range of operating conditions. The robust PSS has been tested ona single machine and on a multimachine system. ( 1997 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth., Vol. 18, 179—193 (1997)(No. of Figures: 16 No. of Tables: 0 No. of Refs: 10)

KEY WORDS control theory; power system stabilizer; robust control

1. INTRODUCTION

Power system stabilizers (PSSs) have been widely used in electric power systems to provide extradamping for synchronous generators. Studies have shown that this is an effective way to improvepower system dynamic stability.1 Over the past two decades, various control methods have beenproposed for PSS design to improve system performance. Among these, conventional lead—lagcompensation PSS2 is adopted by most utility companies because of its simple structure and easeof implementation.

One of the most important features of electric power systems is that they constantly experiencechanges in generation, transmission and load conditions. The conventional linear control designmethod for a PSS requires the nominal structural model to be formulated as a linear, time-invariant system. Changes in the working environment bring discrepancies between the math-ematical description of the power system and the real physical structure. Even under the nominaloperating condition there is still some uncertainty present owing to the approximate knowledgeof the power system parameters, neglected high-frequency dynamics and invalid assumptionsmade in the model formulation process. The conventional PSS is designed based on a nominaloperating point and does give adequate performance for the condition. Enhanced damping isused to indirectly provide a certain degree of tolerance to system uncertainty. However, one of themajor disadvantages of the conventional PSS is that it does not always guarantee system stabilitywith varying operating conditions.3

*Correspondence to: O. P. Malik, Department of Electrical and Computer Engineering, The University of Calgary,2500 University Drive N.W., Calgary, Alta. T2N 1N4, Canada.

CCC 0143—2087/97/030179—15$17.50 Received 22 July 1993( 1997 by John Wiley & Sons, Ltd. Revised 19 June 1996

Electric power systems also suffer from exogenous disturbances such as changes in load,variations in input mechanical power, etc. The a priori information about these external distur-bances is in the form of a certain frequency band in which their energy is concentrated rather thana white-noise-type disturbance. White noise is so far the most popular noise model in the controlliterature because it is easy to handle using stochastic theory, but this way of modelling thedisturbances is not the most appropriate for processes such as power systems. Hence it isdesirable to develop a PSS which can reject external disturbances other than white noise and atthe same time ensure system stability even under system uncertainty.

The H=

optimization method4 is an ideal tool to deal with the robust control design problem.The resulting controller minimizes the effect of external disturbances on the system output interms of the H

=norm, which can easily put the various types of disturbances into a single

framework by using a frequency-weighting function to emphasize the interesting noise band. Themismatch between the physical system and its mathematical description has been taken intoaccount in the control design process to cope with system uncertainty. This paper focuses on thedesign of control laws for a robust power system stabilizer and incorporates the mismatchbetween the physical system and its mathematical description into the control design process.The robust controller has been tested in both the frequency domain and the time domain.Results show that the robust PSS obtained using an H

=design technique satisfies the design

specifications.

2. SYSTEM STRUCTURE AND DESIGN MODEL

A simplified schematic diagram of a single-machine infinite-bus system, considered as a prototypefor robust power system control design, is shown in Figure 1.

The system consists of a generating unit connected to a constant-voltage bus through twoparallel transmission lines. An excitation system and automatic voltage regulator (AVR) areemployed to control the terminal voltage. An associated governor monitors the shaft frequencyand controls the mechanical power. The system representation in the d—q frame of reference5 isgiven in the Appendix. In order to reduce the controller order, which is equal to the order of thesystem plus the order of the weighting functions used in the control design procedure, it isdesirable to have a lower-order system description.

Figure 1. Power system configuration

180 S. CHEN, O. P. MALIK AND T. CHEN

Optim. Control Appl. Meth., Vol. 18, 179—193 (1997) ( 1997 by John Wiley & Sons, Ltd.

Neglecting the transients in the stator circuit and the effect of rotor amortisseur, simplifiedsynchronous generator equations5 are given below in the form of Park’s two-axis machinerepresentation:

d0 "u0u (1)

Mu5 "!k$u#¹

.!¹

%(2)

¹@$0

eR @2"v

&!(x

$!x@

$)i$!e@

2(3)

0"v$#r

!i$!x

2i2

(4)

e@2"v

2#r

!i2#x@

$i$

(5)

¹%"e@

2i2!(x@

$!x

2) i$i2

(6)

v25"v2

$#v2

2(7)

where d denotes the machine load angle in radians, u is the machine rotor speed in p.u. (per unit)and e@

2represents the quadrature axis transient voltage in p.u. The transmission network with

impedance r%#jx

%, which is connected to an infinite bus with voltage v

", is represented by the

equations

v$"v

"sin d#r

%i$!x

%i2

(8)

v2"v

"cos d#r

%i2!x

%i$

(9)

where v$, v

2, i

$and i

2represent direct and quadrature axis voltages and currents respectively.

A simplified AVR and excitation system model is given in the form of a first-order filter as

¹!eR&"K

!(v

3%&#v

4!v

5) (10)

where e&is the field voltage, v

5is the terminal voltage, v

4is a supplementary control signal and v

3%&is the reference input voltage. Under normal operating conditions a linear, time-invariant systemcan be derived by applying small-perturbation relations around a certain equilibrium point. Thestate space notation of the linearized system can be written as

*x5 "A0*x#B

0*u#G

0*w (11)

*y"C0*x (12)

where the system state variable x"[*d *u *e@2

*e&]T, the system output signal y"*u, the

external disturbance w"[*¹.

*»3%&

]T and the system control *u"*v4. The elements of the

constant system matrix A0, input matrix B

0, disturbance input matrix G

0and output matrix C

0are functions of the system parameters and nominal operating parameters.

It is obvious that this linearized system model is a function of the network configuration,system loading condition, etc. Hence the prespecified system parameters will shift from theirnominal values as the operating conditions vary. A serious situation can also emerge duringmajor disturbances even if the system is normally operated. The conventional linear PSS designmethod may exhibit poor performance and cannot retain overall system stability over a widerange of operating conditions.

3. ROBUST PSS DESIGN

In this section, H=

robust control theory is applied to design a robust power system stabilizer. H=

theory provides a powerful tool to address the full range of system stabilization and disturbance

ROBUST POWER SYSTEM STABILIZER 181

( 1997 by John Wiley & Sons, Ltd. Optim. Control Appl. Meth., Vol. 18, 179—193 (1997)

Figure 2. Standard feedback configuration

attenuation problems. The design law takes into account the external disturbances and un-modelled perturbances of the system and formulates them within a single H

=optimal control

framework.4

3.1. Overview of H=

control theory

The basic feedback control system configuration presented in Figure 2 is used here to illustratethe fundamental concept of H

=optimal control theory.

A controller K is employed to reduce the effect of a disturbance d on the output y and at thesame time to enable the system to be internally stable with the system uncertainty.

There are always some uncertainties present in system modelling owing to incomplete know-ledge of the physical parameters, neglected high-frequency dynamics and invalid assumptionsmade in model formulation. The discrepancies between the mathematical model used to describethe system and the real structure of the system can generally be approximated as a norm-bounded, frequency domain uncertainty *G around the nominal plant G

0in the form G(s)"

G0(s)#*G(s), where the perturbation *G is bounded by a frequency domain function R (s) in the

sense that ER~1*GE=)1. Therefore R is considered as an upper bound of the system uncertain-

ty. The condition for robust stabilization given in the form of H=

is

ERK(1#GK)~1E=)1 (13)

In industrial processes the a priori information about external disturbances is always in theform of a certain frequency band in which their energy is concentrated. These kinds of distur-bances can be expressed as a disturbance signal class d weighted by a function ¼(s) such that theH

2norm E¼~1dE

2)1, with emphasis on a certain frequency band in which the disturbance is

most likely to be rejected. The disturbance attenuation objective can be stated as ‘minimize theoutput y for the worst d in the above class’. This is also equivalent to minimizing the H

=norm of

the weighted sensitivity function:

min E¼ (1#GK)~1E=

(14)

It is obvious that the control objectives, i.e. robust stability and disturbance attenuation, can beachieved if the above H

=norm bounds are achieved. The complete solution of the H

=optimiza-

tion problem is given in the state space form with observer structure.6

3.2. Uncertainty description

The power system model presented in Section 2 lacks some information about systemnon-linearities, high-frequency dynamics and unmodelled dynamics which are due to the varyingoperating conditions and assumptions used in model derivation. Frequency domain weightingfunctions are selected as norm-bounded perturbations to represent system uncertainty.The description is simplified by lumping various physical uncertainty sources into a single



Figure 3. Uncertainty model

perturbation for ease of computation. In robust PSS design the uncertainty is considered asa multiplicative perturbation at the plant input as shown in Figure 3.

In this figure the nominal system transfer function G0(s)"C

0(s) [sI!A

0(s)]~1B

0(s).

The real structure of the system is modelled as G(s)"[1#*¼6(s)]G

0(s) to represent varia-

tions around the nominal working condition and * is normalized to E*E=)1. The weighting

function ¼6(s) is selected as

¼6"5·8

s#3·14

s#50(15)

to describe the upper bound of these perturbations. Perturbation studies based on the syn-chronous machine dynamics with varying operation show that the given weighting function isa realistic modelling of the system uncertainty which covers most of the system normal operatingpoints and even some of the extreme operating conditions.

3.3. Controller design

The performance specifications of the closed-loop system will be evaluated using the weightedH

=norm of transfer functions from disturbances to the system-controlled output given in

equation (14). The weighting functions ¼51

and ¼71

are used to normalize the input disturbancesv3%&

and ¹., while ¼

1is the performance index to specify the disturbance attenuation level. In this

problem they are simply chosen as

¼71"¼

51"0·3 (16)

¼1"0·07 (17)

to reflect a certain disturbance level desired to be rejected over the full frequency range. Thefrequency response of these weighting functions is shown in Figure 4.

Both uncertainty weight and performance weight play important roles in robustness andperformance trade-offs in robust control design. The gain of both weighting functions is treated asa design parameter at the beginning of the robust control design procedure. A desired value isobtained after carefully evaluating the induced controller which should balance well the robust-ness and performance of the closed-loop system.

A control design model of robust PSS is developed to fit into the standard configuration7 of H=

analysis and synthesis shown in Figure 5, where P is the interconnected system which consists ofthe nominal plant G

0, the uncertainty weighting function ¼

6and the performance weighting

functions ¼51, ¼

71and ¼

1. The design problem is then to choose a controller K that minimizes

the H=

norm of the transfer function from the disturbance to the desired out, i.e.

min E¼1(1#G

0K)~1¼

$E=

(18)



Figure 4. Uncertainty and performance weights

Figure 5. General interconnection structure

and at the same time makes the closed-loop system stable over all the bounded uncertainties. Inequation (18), ¼

$is the combination of the two weighting functions ¼

51and ¼

71.

A robust control law is synthesized using the two-Riccati-equation approach.8 The resultingcontroller is given in state space notation form as



A,"C

!7·521E#01 6·891E#01 !4·609E#01 7·816E#00 !6·574E#02

!3·101E#01 1·795E#01 !1·370E#01 2·324E#00 !5.297E#02

3·121E#01 !2·494E#01 1·718E#01 !2·560E#00 4·024E#02

5·730E#00 !4·578E#00 2·827E#00 !1·624E#00 6·825E#01

!9·016E!04 1·089E!02 1·954E!05 !2·001E!05 !4·168E#00 D(19)

B,"C

2·360E#02

7·066E#01

!9·776E#01

!1·795E#01

!0·000E#00 D (20)

C,"[!2·242E!04 2·709E!03 4·862E!06 !4·977E!06 1·140E#01] (21)

The order of the resulting controller is fairly low and easy to implement with advanced digitalcontrol techniques. The robust controller is then built up in the form of a transfer function

K (s)"C,(sI!A

,)~1B

,(22)

4. APPLICATION TO A SINGLE-MACHINE INFINITE-BUS SYSTEM

Simulation studies have been carried out to evaluate the performance of the proposed robustpower system stabilizer (RPSS). A detailed non-linear mathematical model is established in theAppendix to simulate the dynamic behaviour of the single-machine infinite-bus power system asalready shown in Figure 1. Time responses of the system with the RPSS are compared with thosewith an IEEE proposed standard conventional type PSS1A power system stabilizer (CPSS)9whose transfer function is also given in the Appendix. The suggested parameters of the CPSS arecarefully tuned to yield the desired performance under the normal system operating condition.

4.1. Light load test

With the system operating under a light load condition of 0·25 p.u. power with 0·9 power factorlag, a 0·25 p.u. step increase in the input torque reference is applied at time 1 s. The applieddisturbance is large enough to cause the system to work in a non-linear region. The time responseof the rotor speed deviation of the system with the RPSS and CPSS under the above conditions isshown in Figure 6. The response of the generator with both the RPSS and CPSS implies thatthere exists a pair of dominant complex conjugate roots. However, the RPSS can provide extradamping to make the system return to its operating point smoothly in less time than with theCPSS.

4.2. Leading power factor operation test

Sometimes the generator in a power system will work at a leading power factor condition inorder to absorb the capacitive charging current in a high-voltage system. In this case a stringentcontroller is necessary to guarantee stable operation of the generator, because the stability marginof the generator is reduced.



Figure 6. Time response to disturbance under light load condition

Figure 7. Time response to disturbance under leading power factor condition

A leading power factor operation test with the generator operating at a power of 0·3 p.u. withleading power factor 0·9 is conducted. At the time 1 s a 0·1 p.u. step increase in the torquereference is applied to the system. The time response of the system with the RPSS and CPSS isshown in Figure 7. It is obvious that the RPSS provides an effective way to improve the generatorstability margin under the leading power factor operating condition.



Figure 8. Time response under three-phase-to-ground fault

4.3. Fault test

A fault test is applied to verify the performance of the proposed robust power system stabilizerunder transient conditions. The response of the system with the RPSS and CPSS to a majorperturbation of a 200 ms three-phase-to-ground short-circuit at the middle of one transmissionline is shown in Figure 8. The fault is applied at 1 s and the transmission line is successfullyreclosed after 6 s. In this case there exists a fairly large variation which shifts the operation of thesystem from its nominal point. It can be seen that the RPSS exhibits good damping properties.

4.4. Frequency domain robustness test

To test the robust stability of the RPSS and CPSS, the CPSS is inserted into the interconnectedsystem configuration shown in Figure 5 with the same uncertainty, disturbance and performanceweighting functions as those of the RPSS. The frequency domain test is undertaken within thefrequency band of interest by using equation (13) for the robust stability condition and the resultsfor both the RPSS and CPSS are shown in Figure 9. The system with the RPSS achieves robuststability. This conclusion stems from the singular value plot of the nominal weighted inputcomplementary sensitivity function, which has a peak value of about 1·0. On the other hand, thesystem with the CPSS does not achieve robust stability, because the singular value plot peaksat a value of 3·2 at a frequency around 10·0 rad~1. This means that there may exist a perturbance*, such that E*E"1/3·2 (i.e. less than the normalized upper bound of uncertainty, 1.0),which makes the system valid for the robust stability condition under the selected uncertaintydescription.

5. APPLICATION TO A MULTIMACHINE POWER SYSTEM

The proposed stabilizer has been applied to a five-machine power system (Figure 10) that exhibitsmultimode oscillations. In this system, generators d2, d3 and d5 form one area and generators



Figure 9. Robust stability test: RPSS versus CPSS

Figure 10. Network topology of five-machine power system

d1 and d4 form a second area. Generators d3 and d5 are very small compared with the othergenerators.

The two areas are connected through a long transmission line 6—7. Each area normally servesits own load and is almost fully loaded, with a small load flow over the tie-line 6—7. The operatingconditions and parameters for all generating units, transmission lines and loads are given inReference 9.

The RPSS for each individual subsystem is designed based on the decentralized controlstrategy that ‘the dynamic behaviour mainly depends on the generator and its local bus’. Theeffect of the external system on the subsystem is formulated into the bounded frequency domainuncertainty which is added to the subsystem dynamics to provide information about the worstsystem variation with the existence of the external system.



5.1. Step change in torque

A 0·1 p.u. step decrease in the mechanical input torque reference of generator d3 followed bya return to the original operating condition after 9 s results in 1·3 Hz local mode oscillationsbetween generators d2 and d3 and 0·65 Hz inter-area mode oscillations between generators d1and d2 as shown in Figure 11.

With the same disturbance as for Figure 11 the local mode oscillations are damped outeffectively when the RPSS is installed on generator d3 as shown in Figure 12, but the RPSS haslittle effect on the inter-area mode oscillations, because the rated capacity of generator d3 ismuch smaller than that of generators d1 and d2.

The time response with RPSSs installed on generators d1, d2 and d3 is shown in Figure 13.As can be seen, both modes of oscillations are damped out very effectively.

5.2. Three-phase-to-ground fault

A three-phase-to-ground fault is applied at the middle of one transmission line betweenbuses d1 and d6 at time 1 s and cleared 100 ms later by opening the faulted line. At 10 sthe faulted transmission line is restored successfully. The time response of the system dueto this disturbance without any PSS and with RPSSs installed on generators d1, d2and d3 is shown in Figure 14. It can be seen that the RPSSs provide very good system transientperformance.

Figure 11. Multimode oscillations in interconnected power system



Figure 12. System response with RPSS installed on generator d3

Figure 13. System response with three RPSSs installed on generators d1, d2 and d3

6. CONCLUSIONS

The robust power system stabilizer design example considered here has demonstrated thecapability of H

=theory to meet both the stability margin and disturbance attenuation require-

ments expressed in terms of the H=

norm of the specified system transfer function. The design isaccomplished by carefully choosing both the uncertainty weighting function and performanceweighting function. Compared with the conventional power system stabilizer, the robust power



Figure 14. Time response of power system under three-phase-to-ground fault

system stabilizer design approach is superior in terms of the readily achievable closed-loop systemperformance. Simulation results show that the proposed robust power system stabilizer canprovide good damping for synchronous generators over a wide range of operating conditions.

APPENDIX

1. Generator unit:

d0 "u0u

u5 "1

H(¹

.#g#K

$d0 !¹

%)

j0$"e

$#r

!i$#u

0(u#1)j

2

j02"e

2#r

!i2#u

0(u#1)j

$

jQ&"e

&!r

&i&

j0,$"!r

,$i,$

j0,2"!r

,2i,2

2. Transmission network:

v$"v

"sin d#r

%i$!x

%i2

v2"v

"cos d#r

%i2#x

%i$



Figure 15. IEEE standard type ST1A AVR and excitation model

Figure 16. IEEE standard PSS1A type CPSS

3. The IEEE standard type ST1A AVR and excitation model are given in Figure 15.

4. Governor transfer function:

g"Aa#b

1#s¹'B d0

5. The IEEE standard PSS1A type conventional PSS10 is given in Figure 16.6. Parameters in simulation studies:

r!"0·007, r

&"0·00089, x

&"1·330

x$"1·24, x

2"0·743, x

.$"1·126, x

.2"0·6262

r,$"0·023, r

,2"0·023, x

,$"1·15, x

,2"0·625

H"4·0, K$"0·0

r%"0·06, x

%"0·3

RC"0·0, X

C"0·0, K

C"0·08

¹C"0·1, ¹

B"0·03, ¹

C1"0·0, ¹

B1"0·0

¹F"1·0, K

F"0·05, ¹

A"0·01, K

A"200·00



»IMAX

"999, »IMIN

"!999, »AMAX

"999, »AMIN

"!999

»RMAX

"7·8, »RMIN

"!6·7, »OEL

"999, »UEL

"!999

a"!0·001328, b"!0·17

¹1"0·1, ¹

2"0·03, ¹

3"0·0, ¹

4"0·0

¹5"2·5, ¹

6"0·005, A

1"0·0, A

2"0·0

K4"5·0, »

STMAX"0·1, »

STMIN"!0·1

All resistances and reactances are in p.u. (per unit) and all time constant in seconds.

REFERENCES

1. de Mello, F. P. and T. F. Laskowski, ‘Concepts of power system dynamic stability’, IEEE ¹rans Power Appar. Syst.,PAS-94, 827—833 (1979).

2. de Mello, F. P. and C. Concordia, ‘Concepts of synchronous machine stability as affected by excitation control’, IEEE¹rans. Power Appar. Syst., PAS-88, 316—329 (1969).

3. Mohamed, M., D. Thorn and E. Hill, ‘Contrast of power system stabilizer performance on hydro and thermal units’,IEEE ¹rans. Power Appar. Syst., PAS-99, 1522—1533 (1980).

4. Francis, B. A., A Course in H=

Control ¹heory, LNCIS Vol. 88, Springer, New York, 1987.5. Anderson, P. M. and A. A. Fouad, Power System Control and Stability, Iowa State University Press, IA, 1977.6. Doyle, J. C., K. Glover, P. P. Khargonekar and B. A. Francis, ‘State-space solution to standard H

2and H

=control

problems, IEEE ¹rans. Automatic Control, AC-34, 831—847 (1989).7. Skogestad, S., M. Morari and J. C. Doyle, Robust control of ill-conditioned plants: high-purity distillation’, IEEE

¹rans. on Automatic Control, AC-33, 1092—1105 (1988).8. Glover, K. and J. C. Doyle, ‘State-space formulae for all stabilizing controllers that satisfy an H

=norm bound and

relations to risk sensitivity’, Syst. Control ¸ett., 11, 167—172 (1988).9. Chen, G. P. and O. P. Malik, ‘Tracking constrained adaptive power system stabilizer’, IEEE Proc. Generat.¹ransmiss.

Distrib., 142, 149—156 (1995).10. IEEE Excitation System Model Working Group, ‘Excitation system models for power system stability studies, Draft

15 for ANSI/IEEE Standard, P421.5/D15, 1990.

.



a robust power system stabilizer design

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